Multi-rate dem with mismatch noise cancellation for digitally-controlled oscillators

ABSTRACT

A digital fractional-N phase locked loop (PLL) with multi-rate dynamic element matching (DEM) and an adaptive mismatch-noise cancellation (MNC) is provided. The PLL includes a phase error to digital converter and a digital loop filter to suppress quantization noise of the phase error to digital converter and drive a digitally controlled oscillator. A digitally controlled oscillator (DCO) with a multi-rate DEM encoder includes an integer bank of frequency control elements (FCE) and a fractional bank of frequency control elements. Adaptive mismatch-noise cancellation logic operates to cancel DCO phase error arising from frequency control element (FCE) static and dynamic mismatch error by estimating ideal MNC coefficient values during PLL normal operation, estimating MNC coefficient errors at each sample time, and updating the MNC coefficient values to approach zero (FCE) static and dynamic mismatch error.

PRIORITY CLAIM AND REFERENCE TO RELATED APPLICATION

The application claims priority under 35 U.S.C. § 119 and all applicable statutes and treaties from prior provisional application Ser. No. 62/722,276, which was filed Aug. 24, 2018, and is incorporated by reference herein.

STATEMENT OF GOVERNMENT INTEREST

This invention was made with government support under grant number 1617545 awarded by National Science Foundation. The government has certain rights in the invention.

FIELD

A field of the invention is frequency synthesis. Example applications of the invention are in clock generation, in wired and in wireless communications. A particular application of the invention is in wireless transceivers for the generation of radio frequency (RF) local oscillator signals used to up-convert and down-convert transmitted and received RF signals.

LIST OF ABBREVIATIONS

The following abbreviations are used in the description and are provided here for ease of reference.

CMOS Complementary Metal Oxide Semiconductor

DCO Digitally Controlled Oscillator

DEM Dynamic Element Matching

DLF Digital Loop Filter

FCE Frequency Control Element

IC Integrated Circuit

LC-Based Inductor-Capacitor Based

MNC Mismatch-Noise Cancellation

PEDC Phase-error-to-digital converter

PLL Phase Locked Loop

PSD Power Spectral Density

RF Radio Frequency

SB Digital Switching Block

BACKGROUND

Evolving wireless communication standards place increasingly stringent performance requirements on the frequency synthesizers that generate RF local oscillator signals for up and down conversion in wireless transceivers. Conventional analog fractional-N PLLs with digital delta-sigma (As) modulation are the current standard for such frequency synthesizers because of their excellent noise and spurious tone performance. See, e.g., T. A. Riley, M. A. Copeland, T. A. Kwasniewski, “Delta-sigma modulation in fractional-N frequency synthesis,” IEEE Journal of Solid-State Circuits, vol. 28, no. 5, pp. 553-559, May 1993. Unfortunately, they require high-performance analog charge pumps and large-area analog filters, so the trends of CMOS technology scaling and increasingly dense system-on-chip integration have created an inhospitable environment for them.

Digital fractional-N PLLs have been developed over the last decade to address this problem. See, e.g., C. Hsu, M. Z. Straayer, M. H. Perrott, “A Low-Noise, Wide-BW 3.6 GHz Digital AS Fractional-N Frequency Synthesizer with a Noise-Shaping Time-to-Digital Converter and Quantization Noise Cancellation,” IEEE International Solid-State Circuits Conference, pp. 340-341, February 2008. They avoid large analog loop filters and can tolerate device leakage and low supply voltages which makes them better-suited to highly-scaled CMOS technology than analog PLLs. They are increasingly used in place of analog PLLs as frequency synthesizers. To date, analog PLLs have the best phase error performance, but digital PLLs have the lowest circuit area and are more compatible with highly-scaled CMOS IC technology. Thus, reducing phase error in digital PLLs has been the subject of intensive research and development for over a decade.

A continuing problem in digital PLLs concerns frequency control element (FCE) mismatches. Such FCE mismatches remain a significant source of phase error in high-performance digital PLLs. See, C. Weltin-Wu, E. Familier, and I. Galton, “A linearized model for the design of fractional-N digital PLLs based on dual-mode ring oscillator FDCs,” IEEETrans.Circuits Syst. I, Reg.Papers, vol. 62, no. 8, pp. 2013-2023, August 2015. Prior attempts to address the FCE mismatch problem use an offline calibration technique that requires several minutes to complete. See, O. Eliezer, B. Staszewski, J. Mehta, F. Jabbar, and I. Bashir, “Accurate self-characterization of mismatches in a capacitor array of a digitally-controlled oscillator,” in Proc. IEEE Dallas Circuits Syst. Workshop, October 2010, pp. 17-18; O. Eliezer, B. Staszewski, and S. Vemulapalli, “Digitally controlled oscillator in a 65 nm GSM/EDGE transceiver with built-in compensation for capacitor mismatches,” in Proc. IEEE Radio Freq. Integr. Circuits Symp., June 2011, pp. 5-7.

A digitally controlled oscillator (DCO) is an oscillator whose frequency is controlled by one or more FCEs, each of which is controlled by a 1-bit digital sequence. For instance, each FCE in an LC-based DCO contributes to the DCO's tank a capacitance that takes on one of two values depending on the state of the FCE's input bit. Changing the FCE's input bit increases or decreases the DCO frequency by a fixed frequency step.

The instantaneous frequency of a DCO is given by a fixed offset frequency plus f_(tune)(t), where:

$\begin{matrix} {{{f_{tune}(t)} = {\sum\limits_{i = 1}^{N_{FCE}}{f_{i}(t)}}},} & (1) \end{matrix}$

N_(FCE) is the number of FCEs in the DCO, and f_(i)(t) is the contribution of the ith FCE to the DCO frequency. Ideally,

f _(i)(t)=(b _(i)[m _(t)]−½)Δ_(i)  (2)

where b_(i)[m] is the FCE's input bit value (either 0 or 1) over the mth clock interval, m_(t)=└f_(FCE)t┘, f_(FCE) is the clock-rate of the input bit, and Δ_(i) is the FCE's frequency step size.

The DCO's input sequence, d[n], represents the ideal value of f_(tune)(t) over the nth clock interval. For example, suppose d[n] is represented as a 16-bit two's complement code where the least significant bit (LSB) represents a DCO frequency step of Δ (e.g., Δ=100 Hz). Then

$\begin{matrix} {{{d\lbrack n\rbrack} = {\left( {{{- 2^{15}}{d_{15}\lbrack n\rbrack}} + {\sum\limits_{i = 0}^{14}{2^{i}{d_{i}\lbrack n\rbrack}}}} \right)\Delta}},} & (3) \end{matrix}$

where d_(i)[n], for each i=0, 1, . . . , 15, is the value of the ith bit of the code (either 0 or 1) over the nth clock interval.

Ideally, f_(tune)(t)=d[n_(t)], where n_(t)=└f_(in)t┘ and f_(in) is the clock-rate of the DCO input. Equations (1)-(3) with f_(FCE)=f_(in) imply that this can be achieved with a bank of 16 FCEs, where the ith FCE's frequency step size is Δ_(i)=2^(i−1)Δ, b_(i)[n]=d_(i−1)[n] for i=1, 2, . . . , 15, and b₁₆[n]=1−d₁₅[n].

Unfortunately, in PLL applications that require low phase noise, such as local oscillator synthesis for cellular telephone transceivers, DCOs with minimum frequency steps of tens of Hz are required, but most existing FCEs have minimum frequency steps of tens of kHz or more. A common solution to this problem is described next for an example case in which f_(tune)(t) needs to be controlled in steps of Δ, yet the smallest realizable FCE frequency step size is Δ_(min)=2⁸Δ. In this case, the 8 LSBs of d[n] are said to represent the fractional part of d[n] because they cause DCO frequency steps that are fractions of Δ_(min), and the 8 most significant bits (MSBs) of d[n] are said to represent the integer part of d[n] because they cause DCO frequency steps that are multiples of Δ_(min).

The basic approach utilizes two FCE banks: an integer FCE bank controlled by the integer part of d[n], and a fractional FCE bank controlled by the output of an oversampling digital ΔΣ modulator driven by the fractional part of d[n] [Error! Bookmark not defined.]. The ΔΣ modulator's highpass-shaped quantization noise is lowpass filtered by the DCO, so provided the oversampling rate is sufficiently high, it negligibly contributes to the DCO's phase error.

FIG. 1 shows a specific example in the context of an LC-based DCO, where p_(t)=└f_(fast)t┘, f_(fast)>>f_(it), and d_(I)[n_(t)] and d_(F)[n_(t)] are the integer and fractional parts of d[n_(t)], respectively. The f_(fast)-clk signal is such that p_(t) changes synchronously with n_(t), so that n_(t) can be written as a function of p_(t), i.e.,

n _(t) =g(p _(t)).  (4)

In this example g(p_(t))=└(f_(in)/f_(fast))p_(t)┘, where f_(fast)/f_(in) is an integer much greater than 1.

It follows from (3) that d[n_(t)]=d_(I)[n_(t)]+d_(F)[n_(t)], where

$\begin{matrix} {{d_{I}\left\lbrack n_{t} \right\rbrack} = {\left( {{{- 2^{15}}{d_{15}\left\lbrack n_{t} \right\rbrack}} + {\sum\limits_{i = 8}^{14}{2^{i}{d_{i}\left\lbrack n_{t} \right\rbrack}}}} \right)\Delta}} & (5) \\ {and} & \; \\ {{d_{F}\left\lbrack n_{t} \right\rbrack} = {\Delta {\sum\limits_{i = 0}^{7}{2^{i}{{d_{i}\left\lbrack n_{t} \right\rbrack}.}}}}} & (6) \end{matrix}$

As shown in FIG. 1, d_(F)[n_(t)] is sampled at a rate of f_(fast) by a second-order digital ΔΣ modulator. The ΔΣ modulator's output is a four-level sequence quantized to multiples of Δ_(min) and can be written as

y _(ΔΣ)[p _(t)]=d _(F)[n _(t)]+e _(ΔΣ)[p _(t)],  (7)

where e_(ΔΣ)[p_(t)] is second-order highpass-shaped quantization noise plus any dither used within the ΔΣ modulator. A thermometer encoder maps y_(ΔΣ)[p_(t)] to a 4-bit thermometer code which drives a bank of four FCEs, each with a frequency step of Δ_(min). It follows from (1), (2) and (7) that the contribution of the fractional FCE bank to the DCO frequency, f_(F)(t), is

$\begin{matrix} {{f_{F}(t)} = {{\sum\limits_{i = 1}^{4}{f_{i}(t)}} = {{d_{F}\left\lbrack n_{t\;} \right\rbrack} + {{e_{\Delta\Sigma}\left\lbrack p_{t} \right\rbrack}.}}}} & (8) \end{matrix}$

The integer FCE bank is directly driven by d_(I)[n_(t)]. Specifically, the ith FCE, for i=5, 6, . . . , 11, has input b_(i)[n_(t)]=d_(i+3)[n_(t)] and frequency step size Δ_(i)=2^(i+3)Δ, and the 12th FCE has input b₁₂[n_(t)]=1−d₁₅[n_(t)] and frequency step size Δ₁₂=2¹⁵Δ. It follows from (1), (2) and (5) that the contribution of the integer FCE bank to the DCO frequency, f_(I)(t), is

$\begin{matrix} {{{f_{I}(t)} = {{\sum\limits_{i = 5}^{12}{f_{i}(t)}} = {d_{I}\left\lbrack n_{t} \right\rbrack}}},} & (9) \end{matrix}$

where a constant additive term has been omitted.

The contribution of the two FCE banks to the DCO frequency is f_(tune)(t)=f_(I)(t)+f_(F)(t), so (8) and (9) imply that

f _(tune)(t)=d[_(t)]+e _(ΔΣ)[p _(t)]  (10)

Accordingly, e_(ΔΣ)[p_(t)] causes DCO frequency error. The DCO's phase error is the integral of its frequency error, so as mentioned above, a lowpass-filtered version of e_(ΔΣ)[p_(t)] appears as a component of the DCO's phase error. Given that e_(ΔΣ)[p_(t)] has a highpass-shaped spectrum that peaks at f_(fast)/2, its contribution to the DCO's phase error can be made negligible relative to other sources of phase error if f_(fast) is large enough.

The analysis above presumes that the FCEs are ideal. Unfortunately, non-ideal circuit behavior causes f_(i)(t) to deviate from (2). For example, suppose for now that f_(i)(t) is modeled as ideal except for a static gain error given by α_(i), i.e.

f _(i)(t)=(b _(i)[m _(t)]−½)α_(i)Δ_(i).  (11)

Ideally, α_(i)=1 for i=1, 2, . . . , N_(FCE), but inevitable component mismatches introduced during fabrication cause α_(i) to deviate from 1.

Repeating the analysis for the example in FIG. 1 with (11) in place of (2) gives

f _(tune)(t)=α_(F) f _(tune-ideal)(t)+e _(F)(t)+e _(I)(t)+(α_(I)−α_(F))d _(I)[n _(t)],   (12)

where f_(tune-ideal)(t) is given by the right side of (10), α_(F) and α_(I) are the averages of α_(i) for i=1, 2, 3, 4 and i=5, 6, . . . , 12, respectively,

$\begin{matrix} {{e_{F}(t)} = {\sum\limits_{i = 1}^{4}{\left( {\alpha_{i} - \alpha_{F}} \right)\left( {{b_{i}\left\lbrack p_{t} \right\rbrack} - {1/2}} \right)\Delta_{\min}}}} & (13) \\ {and} & \; \\ {{e_{I}(t)} = {\sum\limits_{i = 5}^{12}{\left( {\alpha_{i} - \alpha_{I}} \right)\left( {{b_{i}\left\lbrack n_{t} \right\rbrack} - {1/2}} \right){\Delta_{i}.}}}} & (14) \end{matrix}$

where f_(tune-ideal)(t) is given by the right side of (10), α_(F) and α_(I) are the averages of α_(i) for i=1, 2, 3, 4 and i=5, 6, . . . , 12, respectively.

Hence, the FCE static gain errors introduce a gain factor, α_(F), and three additive error terms to f_(tune)(t). The α_(F) gain factor does not significantly degrade performance in typical PLLs. In contrast, as explained next, the three additive error terms in (12) tend to cause spurious tones and increase phase error in PLLs because they are nonlinear functions of d[n_(t)].

The individual bits of d[n], i.e., d_(i)[n], for each i=0, 1, . . . , 15, each depend on d[n] but are restricted to values of 0 and 1. Hence, each d_(i)[n] is a nonlinear function of d[n]. Nevertheless, they can be combined as in (3) to yield d[n], which implies that multiplying d₀[n], d₁[n], . . . , d₁₄[n], and d₁₅[n] by 2⁰, 2¹, . . . , 2¹⁴, and 2¹⁵, respectively, and adding the results causes the nonlinear components from the individual bits to cancel each other. Any deviation from a set of scale factors proportional to those mentioned above prevents full cancellation of the nonlinear components. It can be verified from (5), (13) and (14) that e_(F)(t), e_(I)(t), and (α_(I)−α_(F))d_(I)[n_(t)] are each a function of a subset of the individual bits of d[n_(t)], so they are nonlinear functions of d[n_(t)].

A partial solution to this problem is to replace the thermometer encoder in FIG. 1 with a mismatch-shaping DEM encoder. See, I. Galton, “Why Dynamic-Element-Matching DACs Work,” IEEE Trans. Circuits Syst. Exp. Briefs, vol. 57, no. 2, pp. 69-74, March 2010. Doing so would cause e_(F)(t) to be replaced by highpass-shaped noise that is free of nonlinear distortion and is uncorrelated with d[n_(t)], so it would be suppressed by the DCO 102 like the ΔΣ quantization noise. Similarly, the integer FCE bank 104 could be modified to accommodate a mismatch-shaping DEM encoder clocked at a rate of f_(in), which would cause e_(I)(t) to be replaced by shaped noise that is free of nonlinear distortion and is uncorrelated with d[n_(t)]. However, f_(in □) f_(fast), less of the shaped noise would be suppressed by the DCO 102. Unfortunately, DEM as described above would not help prevent the last term in (12) from introducing nonlinear distortion because d_(I)[n_(t)] is a non-linear function of d[n_(t)].

The last two terms in (12) increase the phase error in a PLL unless d_(I)[n_(t)] remains constant once the PLL is locked. See, C. Weltin-Wu, G. Zhao, and I. Galton, “A Highly-Digital Frequency Synthesizer Using Ring-Oscillator Frequency-to-Digital Conversion and Noise Cancellation,” IEEE International Solid-State Circuits Conf., pp. 1-3, February 2015. In most published digital PLLs d[n] varies by much less than Δ_(min) when the PLL is locked, and measured results are usually presented for PLL frequencies at which can d_(I)[n_(t)] does not change during the measurement interval. This renders the last two terms in (12) constant, so they do not contribute phase error. Unfortunately, this is not a viable option in practice because DCO center frequency drift caused by flicker noise, voltage and temperature variations, and pulling from external interference cause d[n_(t)] to vary by far more than Δ_(min) over time. For instance, measurement results indicate that the frequency of the DCO presented in [C. Weltin-Wu, G. Zhao, and I. Galton, “A Highly-Digital Frequency Synthesizer Using Ring-Oscillator Frequency-to-Digital Conversion and Noise Cancellation,” IEEE International Solid-State Circuits Conf., pp. 1-3, February 2015] varies by about −200 kHz/° C., which corresponds to ˜7Δ_(min) per degree Celsius. In practice, this causes the digital PLL's phase noise to increase drastically from time to time as d[n_(t)] slowly drifts past integer multiples of Δ_(min). This issue is sometimes called “spectral breathing” because the phase noise spectrum, as viewed on laboratory measurement equipment, appears to swell up every now and then as if it is taking deep breaths. During these “breaths” the PLL's performance is extremely degraded. Furthermore, when the PLL is used to generate phase or frequency modulated signals, such as a GFSK signal for a Bluetooth transmitter, d[n_(t)] typically varies by more than Δ_(min), so there are no periods between “breaths” during which the phase noise performance is good.

To address this problem, a single bank of FCEs driven by a ΔΣ modulator and a mismatch-shaping DEM encoder could be used, where the ΔΣ modulator oversamples d[n_(t)] instead of just d_(F)[n_(t)]. The DEM encoder would cause any mismatches among the FCEs to contribute shaped noise instead of nonlinear distortion, and the oversampling would ensure that most of the noise is suppressed by the DCO. Unfortunately, high oversampling ratios would be required in practice, which makes this solution impractical because of the associated high-power consumption

SUMMARY OF THE INVENTION

A preferred embodiment is a digital fractional-N phase locked loop (PLL) with multi-rate dynamic element matching (DEM) and an adaptive mismatch-noise cancellation (MNC). The PLL includes a phase error to digital converter (PEDC) and a digital loop filter to suppress quantization noise of the PEDC and drive a digitally controlled oscillator. A digitally controlled oscillator (DCO) with a multi-rate DEM encoder includes an integer bank of frequency control elements (FCE) and a fractional bank of frequency control elements. Adaptive mismatch-noise cancellation logic operates to cancel DCO phase error arising from frequency control element (FCE) static and dynamic mismatch error by estimating ideal MNC coefficient values during PLL normal operation, estimating MNC coefficient errors at each sample time, and updating the MNC coefficient values to approach zero (FCE) static and dynamic mismatch error.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 (Prior Art) is a schematic diagram of a frequency control technique for an LC-based DC);

FIG. 2 includes a set of example waveforms related to equations (15) and (16) to provide a visual representation of the effects of FCE mismatches on the DCO frequency;

FIG. 3 (Prior Art) is a block diagram of a mismatch-shaping segmented DEM encoder;

FIGS. 4A-4C (Prior Art) are block (4A & 4B) and schematic (4C) diagrams showing digital switching blocks of the FIG. 3 DEM encoder;

FIG. 5 is a block diagram of a preferred embodiment DCO with a multi-rate segmented DEM encoder that is a modification of the FIG. 3 encoder;

FIG. 6 is a block diagram of a preferred slow DEM encoder used in FIG. 5,

FIG. 7 is a functional diagram of a preferred embodiment second-order digital ΔΣ modulator used in FIG. 5;

FIG. 8 shows the fractional path of the multi-rate DEM encoder shown in FIG. 5 modified to accommodate MNC;

FIG. 9 (Prior Art) is a schematic diagram of a digital fractional-N PLL without MNC;

FIGS. 10A & 10B (Prior Art) respectively are a schematic diagram of a synchronization circuit used at DCO input in FIG. 9 and illustration of the clock signals within the DCO, p_(t), and n_(t)=g(p_(t)) for f_(fast)=4. 5f_(ref),

FIGS. 11A-11C are block diagrams that illustrate a preferred digital fractional-N PLL with the multi-rate DEM encoder and MNC;

FIG. 12 shows example frequency transients used in a simulation of the FIGS. 11A-11C digital fractional-N PLL;

FIGS. 13A-13C respectively show simulated phase noise of the FIGS. 11A-11C digital fractional-N PLL; with the multi-rate DEM technique disabled, with the multi-rate DEM technique enabled for two cases, and with the multi-rate DEM technique enabled and with the MNC technique disabled and enabled;

FIGS. 14A & 14B show the evolution of the MNC coefficient errors over time from the simulation used to generate the curves in FIG. 13C;

FIGS. 15A & 15B show the evolution of the MNC coefficient errors over time for 7·8·10⁷ reference periods (3 seconds) for an example case in which K_(a) and K_(b) are initially set to 2⁻¹ and 2⁻², respectively, and then divided by two at the times indicated by the vertical dashed lines.

DESCRIPTION OF PREFERRED EMBODIMENTS

Preferred embodiment methods and digital oscillators provide multi-rate dynamic element matching (DEM) and an adaptive mismatch-noise cancellation (MNC) that work together to address FCE mismatches. The DEM and the MNC run during normal PLL operation, and the MNC typically converges in a few seconds from a cold start. A preferred DEM has been simulated and succeeds in reducing noise from the FCE mismatches. The MNC cancels DCO phase error arising from FCE mismatch error. Ideal MNC coefficient values are estimated, during PLL normal operation, as part of the feedback loop in a digital fractional-N PLL that incorporates the DCO.

The center frequency of a conventional digitally-controlled oscillator (DCO) drifts over time due to flicker noise, voltage and temperature variations, and pulling from external interference. Given that the DCO frequency is a non-linear function of the DCO's input signal, this causes the digital PLL's phase noise to increase drastically from time to time because the DCO's input signal slowly drifts to counteract the DCO's center frequency drift. This issue is called spectral breathing because the phase noise spectrum, as viewed on laboratory measurement equipment, appears to swell up every now and then as if it is taking deep breaths of air, during which the PLL's performance is extremely degraded. Moreover, when the PLL is used to generate phase or frequency modulated signals there are no periods between breaths during which the phase noise performance is good. Spectral breathing can drastically degrade a digital PLL's phase noise. Preferred embodiments address spectral breathing by making the relation between the DCO frequency and its input signal linear, which is done at the expense of initially adding more noise to the system. However, this added noise has properties that can be exploited to cancel it, so that the digital PLL's performance is no longer degraded when the DCO's input signal changes. Overall, the price is only a slightly higher power consumption.

Preferred embodiments provide a new multi-rate DEM technique and an MNC technique that work together within a PLL to solve the problems that arise from FCE mismatches are presented. As in FIG. 1, the preferred embodiment uses integer and fractional FCE banks. In the preferred embodiment, both FCE banks are driven by a multi-rate DEM encoder, which ensures that the error arising from FCE mismatches is free of nonlinear distortion. In addition, the multi-rate DEM encoder avoids high power consumption because most of its digital logic is clocked at a rate of f_(in) instead of f_(fast). Although the hardware of preferred embodiments is different from that of the solution in which d[n_(t)] is oversampled and a DEM encoder clocked at a high rate is used to control the FCEs, a pessimistic power consumption analysis indicates that the preferred techniques are at least five times more power-efficient. Much of the additive error is not oversampled, so instead of relying on the DCO to suppress it, the MNC technique adaptively measures the error and cancels it in real time.

Preferred embodiments of the invention will now be discussed with respect to the drawings and experiments used to demonstrate the invention. The drawings may include schematic and/or block representations, which will be understood by artisans in view of the general knowledge in the art and the description that follows.

FCEs with Δ_(i)>Δ_(min) are usually built by connecting nominally identical minimum-weight FCEs in parallel. Static mismatches among these FCEs are sources of error, but other non-idealities such as the non-instantaneous frequency transitions of realizable FCEs are also sources of error. Hence, a more comprehensive model than (11) for f_(i)(t) is

f _(i)(t)=(b _(i)[m _(t)]−½)Δ_(i) +e _(i)(t),  (15)

where e_(i)(t) is error that models both the static mismatch and the non-ideal frequency transitions of the ith FCE. b_(i)[m] is the FCE's input bit value (either 0 or 1) over the mth clock interval, as defined above in (2). FCEs are designed such that frequency transitions caused by input bit changes settle within a clock period, so e_(i)(t) only depends on b_(i)[m_(t)−1] and b_(i)[m_(t)]. This can be modeled as

$\begin{matrix} {{e_{i}(t)} = \left\{ {\begin{matrix} {e_{11i},} & {{{{if}\mspace{14mu} {b_{i}\left\lbrack {m_{t} - 1} \right\rbrack}} = 1},{{b_{i}\left\lbrack m_{t} \right\rbrack} = 1},} \\ {{e_{01i}(t)},} & {{{{if}\mspace{14mu} {b_{i}\left\lbrack {m_{t} - 1} \right\rbrack}} = 0},{{b_{i}\left\lbrack m_{t} \right\rbrack} = 1},} \\ {e_{00i},} & {{{{if}\mspace{14mu} {b_{i}\left\lbrack {m_{t} - 1} \right\rbrack}} = 0},{{b_{i}\left\lbrack m_{t} \right\rbrack} = 0},} \\ {{e_{10i}(t)},} & {{{{if}\mspace{14mu} {b_{i}\left\lbrack {m_{t} - 1} \right\rbrack}} = 1},{{b_{i}\left\lbrack m_{t} \right\rbrack} = 0}} \end{matrix},} \right.} & (16) \end{matrix}$

where e_(11i), e_(01i)(t), e_(00i), and e_(10i)(t) represent the error over each clock interval corresponding to the four different possibilities of the FCE's current and prior input bit values. The FCE model given by (15) and (16) is analogous to that of a non-return-to-zero (NRZ) 1-bit DAC. To prevent e_(i)(t) from depending on b_(i)[m_(t)−1], return-to-zero (RZ) FCEs could be implemented by setting the FCEs to a signal-independent state for a fraction of each clock period, but this is not practical for PLLs because it would periodically slew the DCO frequency and thereby introduce excessive phase noise.

FIG. 2 shows example waveforms associated with (15) and (16). A consequence of the frequency transitions settling within a clock period is that when an FCE's input bit does not change between clock periods, neither does its contribution to the DCO frequency, so e_(00i) and e_(11i) are constant. In contrast, e_(01i)(t) and e_(10i)(t) are not constant because they represent deviations from the FCE's ideal instantaneous frequency transitions when its input bit changes. As shown in FIG. 2, the shape of each of these frequency transitions depends only on whether the corresponding FCE input changed from 0 to 1 or 1 to 0, and both e_(01i)(t) and e_(10i)(t) are 1/f_(FCE)-periodic.

Experimental results indicate, at least for the LC-based DCOs presented in [C. Venerus and I. Galton, “A TDC-Free Mostly-Digital FDC-PLL Frequency Synthesizer with a 2.8-3.5 GHz DCO,” IEEE J. Solid-State Circuits, vol. 50, no. 2, pp. 450-463, February 2015] and [C. Weltin-Wu, E. Familier, and I. Galton, “A Linearized Model for the Design of Fractional-N PLLs based on Dual-Mode Ring Oscillator FDCs,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 8, pp. 2013-2023, August 2015], that the frequency transition introduced by each FCE when its input bit changes from 0 to 1 and that when the input bit changes from 1 to 0 are antisymmetric to a high degree of accuracy, i.e., e_(11i)−e_(01i)(t)=−[e_(00i)−e_(10i)(t)]. Therefore, substituting (16) into (15), applying this observation, collecting terms and omitting constant additive terms yields.

f _(i)(t)=(b _(i)[m _(t)]−½)α_(i)(t)Δ_(i)+(b _(i)[m _(t)−1]−½)γ_(i)(t),  (17)

where

α_(i)(t)=1+(e _(01i)(t)−e _(00i))/Δ_(i)and γ_(i)(t)=e _(11i) −e _(01i)(t).  (18)

Given that α_(i)(t) and γ_(i)(t) are functions of e_(01i)(t) and e_(10i)(t), which are 1/f_(FCE)-periodic, they are also 1/f_(FCE)-periodic.

Multi-Rate DEM

Single-Rate Segmented DEM

Suppose the DCO's input sequence is given by (3), and for now suppose that ΔΣ quantization is not necessary because FCEs with small-enough step sizes are available, i.e., Δ_(min)=Δ. Even in this case, FCE mismatches are a problem because they cause nonlinear distortion. A conventional single-rate segmented DEM encoder can be used to prevent this problem. For example, the mismatch-shaping segmented DEM encoder shown in FIG. 3 can be used with 34 FCEs. See, K. L. Chan, N. Rakuljic, and I. Galton, “Segmented Dynamic Element Matching for High-Resolution Digital-to-Analog Conversion,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 55, no. 11, pp. 3383-3392, December 2008. The ith FCE has input b_(i)[n_(t)]=c_(i)[n_(t)] and frequency step size Δ_(i)=K_(i)Δ, where

K _(2i-1) =K _(2i)=2^(i−1) for i=1,2, . . . ,13, and

K _(i)=2¹³ for i=27,28, . . . ,34.  (19)

The DEM encoder's input sequence, c[n_(t)], is obtained from the DCO input sequence as

c[n _(t)]=d[n _(t)]/Δ+2¹⁵+2¹³−1  (20)

As shown in FIG. 3, the DEM encoder 300 consists of 33 digital switching blocks (SBs) 302, labeled S_(k,r) for k=1, 2, . . . , 16, and r=1, 2, . . . , 17, configured in a tree structure. The 13 shaded SBs are called segmenting SBs, whereas the other 20 SBs are called non-segmenting SBs. The functional details of the SBs are shown in FIGS. 4A-4C. The top and bottom outputs of each segmenting SB are ½(c_(k,r)[n_(t)]−1−s_(k,1)[n_(t)]) and 1+s_(k,1)[n_(t)], respectively, where c_(k,1)[n_(t)] is the SB input sequence, and s_(k,1)[n_(t)], called a switching sequence, is 0 when c_(k,1)[n_(t)] is odd and ±1 otherwise. Similarly, the top and bottom outputs of each non-segmenting SB are ½(c_(k,r)[n_(t)]−s_(k,r)[n_(t)]) and ½(c_(k,r)[n_(t)]+s_(k,r)[n_(t)]), respectively, where c_(k,r)[n_(t)] is the SB input sequence and s_(k,r)[n_(t)] is 0 when c_(k,r)[n_(t)] is even and ±1 otherwise.

Regardless of the SB type, each switching sequence is zero-mean and has a first-order highpass-shaped power spectral density (PSD) that peaks at f_(in)/2. It is generated in two's complement format by the logic shown in FIG. 4C, wherein d_(k,r)[n_(t)] is generated within each SB and is well-modeled as a two-level white random sequence that takes on values of 0 and 1 with equal probability and is independent of the d_(k,r)[n_(t)] sequences in the other SBs.

Extension to Multi-Rate Segmented DEM

Now suppose that the smallest practical FCE frequency step size is Δ_(min)=2⁸Δ. As the lower 16 FCEs in the example above all have frequency step sizes smaller than Δ_(min), the bottom 16 outputs of the DEM encoder can no longer drive FCEs directly. The preferred multi-rate DEM architecture 500 in the DCO control logic 501 shown in FIG. 5 addresses this situation, where a bottom 4 FCEs make up a fractional FCE bank 502, the top 18 FCEs make up an integer FCE bank 504, and w_(t)=p_(t)−1 is a T_(fast)-delayed version of p_(t), where T_(fast)=¹/f_(fast). As in FIG. 1, n_(t)=g(p_(t)) changes synchronously with p_(t).

A slow DEM encoder 506 is a modified version of the DEM encoder in FIG. 3. Its outputs c₁₇[n_(t)], c₁₈[n_(t)], . . . , c₃₄[n_(t)] are identical to those in FIG. 3, and instead of outputs c₁[n_(t)], c₂[n_(t)], . . . , c₁₆[n_(t)] it has an output, x_(f)[n_(t)], supplied to a second order digital ΔΣ modulator 508, given by

$\begin{matrix} {{x_{f}\left\lbrack n_{t} \right\rbrack} = {\Delta {\sum\limits_{i = 1}^{16}{{K_{i}\left( {{c_{i}\left\lbrack n_{t} \right\rbrack} - {1/2}} \right)}.}}}} & (21) \end{matrix}$

Each c_(i)[n_(t)] takes on values of 0 and 1, so (19) and (21) imply that |x_(f)[n_(t)]|≤255Δ and x_(f)[n_(t)] is restricted to multiples of Δ.

The slow DEM encoder could be implemented from the DEM encoder of FIG. 3 directly by combining c₁[n_(t)], c₂[n_(t)], . . . , c₁₆[n_(t)] as in (21), but the preferred structure of FIG. 6 provides a simpler and more elegant approach. As implied by FIG. 4B, the sum of the outputs of each non-segmenting SB is equal to the SB's input, so it follows from (21), FIG. 3 and FIG. 4A that x_(f)[n_(t)] can be computed directly from the bottom outputs of S_(16,1), S_(15,1), . . . , S_(9,1) as

$\begin{matrix} {{x_{f}\left\lbrack n_{t} \right\rbrack} = {\Delta {\sum\limits_{k = 9}^{16}{2^{16 - k}{{s_{k,1}\left\lbrack n_{t} \right\rbrack}.}}}}} & (22) \end{matrix}$

Hence, as shown in FIG. 6, S_(1,1), S_(1,2), . . . , S_(1,8) are not necessary in the slow DEM encoder 506.

The Δ scale factor shown in FIG. 6 is not an actual multiplier; it just denotes that the subsequent digital logic should interpret the LSB of x_(f)[n_(t)] to represent a DCO frequency step size of Δ.

As shown in FIG. 5, x_(f)[n_(t)] is sampled at a rate of f_(fast) by the second-order digital 4E modulator 508, whose functional diagram is shown in FIG. 7. The dither sequence, d_(ΔΣ)[p_(t)], is generated such that it can be well-modeled as a two-level white random sequence that is independent of d[n_(t)] and x_(f)[n_(t)] and takes on values of 0 and A with equal probability. It ensures that the ΔΣ modulator's 508 quantization noise is asymptotically independent of x_(f)[n_(t)] and d_(ΔΣ)[p_(t)], and has a PSD equal to that of the output of a filter with transfer function (1−z⁻¹)² driven by white noise with a variance of Δ_(min) ²/12. See, S. Pamarti, J. Welz, and I. Galton, “Statistics of the Quantization Noise in 1-Bit Dithered Single-Quantizer Digital Delta-Sigma Modulators,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 3, pp. 492-503, March 2007. The ΔΣ modulator output is quantized to values in the set {−2Δ_(min), −Δ_(min), 0, Δ_(min), 2Δ_(min)} and is given by

y _(ΔΣ)[p _(t)]=x _(f)[n _(t)]+e _(ΔΣ)[p _(t)],  (23)

where e_(ΔΣ)[p_(t)] is second-order highpass-shaped quantization noise plus d_(ΔΣ)[p_(t)].

In FIG. 5, a fast DEM encoder 510 can be a conventional mismatch-shaping non-segmented DEM encoder with a clock rate of f_(fast). It is implemented as a tree of non-segmenting SBs, and it maps y_(ΔΣ)[p_(t)] to four 1-bit sequences, each of which drives an FCE with a frequency step size of Δ_(min).

Each b_(i)[w_(t)] in FIG. 5, for i=1, 2, 3, 4, is clocked at a rate of f_(fast) and toggles rapidly enough such that the FCE frequency transitions from the fractional FCE bank introduce high-frequency error components to the DCO phase error. Such components are lowpass filtered by the DCO, so f_(fast) can be chosen so that they are not a problem in practice. Consequently, the frequency transitions of the FCEs from the fractional FCE bank 502 are modeled as ideal, so that f_(i)(t) is given by (11) for i=1, 2, 3, 4.

It follows that

f _(F)(t)=α_(F) y _(ΔΣ)[w _(t)]+e _(F)(t),  (24)

where α_(F) is the average of α_(i) for i=1, 2, 3, 4 and e_(F)(t) is a function of the errors introduced by the fractional FCE bank 502 and the switching sequences from the fast DEM encoder 510. The fast DEM encoder 510 ensures that e_(F)(t) is free of nonlinear distortion, uncorrelated with y_(ΔΣ)[w_(t)], and has a first-order highpass-shaped PSD that peaks at f_(fast)/2, so f_(fast) can be chosen so that this term is not a problem in practice. Thus, substituting (23) into (24) and neglecting e_(F)(t) gives

f _(F)(t)=α_(F) x _(f)[g(w _(t))]+α_(F) e _(ΔΣ)[w _(t)].  (25)

As shown in FIG. 5, the c₁₇[n_(t)], c₁₈[n_(t)], . . . , c₃₄[n_(t)] outputs of the slow DEM encoder 506 drive the same FCEs as those of the DEM encoder of FIG. 3. This implies that f_(I)(t) is given by

$\begin{matrix} {{{f_{I}(t)} = {{{\alpha_{I}(t)}{d\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} + {{\gamma_{I}(t)}{d\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack}} + {e_{I}(t)}}},} & (26) \\ {where} & \; \\ {{{e_{I}(t)} = {\Delta {\sum\limits_{k,r}^{\;}\left\{ {{{\alpha_{k,r}(t)}{s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} + {{\gamma_{k,r}(t)}{s_{k,r}\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack}}} \right\}}}},} & (27) \end{matrix}$

α_(I)(t), γ_(I)(t), α_(k,r)(t) and γ_(k,r)(t) (defined in Appendix A below) are T_(fast)-periodic waveforms that depend on the errors introduced by the integer FCE bank 504, and the summation indices indicate the summation over all k and r values corresponding to the SBs within the slow DEM encoder 506.

The contribution to the DCO frequency from both FCE banks 502 and 504 is f_(tune)(t)=f_(I)(t)+f_(F)(t), so (25) and (26) imply that

f _(tune)(t)=α_(I)(t)d[g(w _(t))]+γ_(I)(t)d[g(w _(t)−1)]+α_(F) e _(ΔΣ)[w _(t)]+e _(M)(t),  (28)

where

e _(M)(t)=e _(I)(t)+α_(F) x _(f)[g(w _(t))]  (29)

is called FCE mismatch error. e_(M)(t) is a linear combination of the switching sequences from the slow DEM encoder whose coefficients depend on the errors introduced by both FCE banks 502 and 504.

The γ_(I)(t)d[g(w_(t)−1)] term in (28) is proportional to a T_(fast)-delayed version of d[g(w_(t))], so it represents a linear filtering operation. This term tends to be much smaller than the desired signal component, α_(I)(t)d[g(w_(t))], so it is not a problem in practice. The α_(F)e_(ΔΣ)[w_(t)] term is proportional to ΔΣ quantization noise plus dither so it is free of nonlinear distortion, is uncorrelated with the other terms in (28), and has a highpass-shaped PSD. The e_(M)(t) term also has these properties because it is a linear combination of the switching sequences from the slow DEM encoder. The PSD of α_(F)e_(ΔΣ)[w_(t)] peaks at f_(fast)/2, whereas the PSD of e_(M)(t) peaks at f_(in)/2. Hence, f_(fast) can be increased to make the DCO phase error introduced by α_(F)e_(ΔΣ)[w_(t)] negligible, but this would not reduce the DCO phase error contribution from e_(M)(t). Therefore, e_(M)(t) is the only problematic term in (28).

Substituting (22) and (27) into (29) yields

$\begin{matrix} {{{e_{M}(t)} = {\Delta {\sum\limits_{k,r}^{\;}\left\{ {{\delta_{k,r}{s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} + {{\gamma_{k,r}(t)}\left( {{s_{k,r}\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack} - {s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} \right)}} \right\}}}},} & (30) \\ {\mspace{79mu} {where}} & \; \\ {\mspace{79mu} {\delta_{k,r} = \left\{ {\begin{matrix} {{{\alpha_{k,r}(t)} + {\gamma_{k,r}(t)} + {\alpha_{F}2^{16 - k}}},} & {{{{if}\mspace{14mu} k} \geq 9},{r = 1}} \\ {{{\alpha_{k,r}(t)} + {\gamma_{k,r}(t)}},} & {otherwise} \end{matrix},} \right.}} & (31) \end{matrix}$

is constant for each k and r, even though neither α_(k,r)(t) nor γ_(k,r)(t) are constant. The non-constant terms in each α_(k,r)(t) are equal in magnitude but opposite in sign to the corresponding terms in γ_(k,r)(t), so α_(k,r)(t)+γ_(k,r)(t), and hence δ_(k,r), are constant. Therefore, the terms proportional to δ_(k,r), in (30) represent the DCO frequency error contribution from FCE static gain errors, whereas the terms proportional to γ_(k,r)(t) in (30) represent the DCO frequency error contribution from non-ideal FCE frequency transitions.

Adaptive FCE Mismatch Noise Cancellation

The purpose of the present MNC is to cancel most of the DCO phase error that would otherwise be caused by e_(M)(t). To do this, the sequence

$\begin{matrix} {{{e_{MNC}\left\lbrack p_{t} \right\rbrack} = {\Delta {\sum\limits_{k,r}^{\;}\left\{ {{\alpha_{k,r}{s_{k,r}\left\lbrack n_{t} \right\rbrack}} + {b_{k,r}\left( {{s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack} - {s_{k,r}\left\lbrack n_{t} \right\rbrack}} \right)}} \right\}}}},} & (32) \end{matrix}$

where a_(k,r), and b_(k,r) are called the MNC coefficients, is injected into the fractional path of the multi-rate DEM encoder. The ideal MNC coefficient values, i.e., the values of a_(k,r), and b_(k,r), for which the DCO phase error contribution of e_(M)(t) is minimized, are estimated with a least-mean-square (LMS)-like algorithm. The algorithm is similar to a conventional LMS algorithm in the sense that it consists of a set of coefficients that are updated over time based on how strongly known signals are correlated to an error measurement.

We next explain how e_(MNC)[p_(t)] affects the DCO's phase error, how the FCE mismatch error is measured, and how the MNC coefficients are adaptively computed from the FCE mismatch error measurement.

MNC Sequence Application

FIG. 8 shows the fractional path of the multi-rate DEM encoder shown in FIG. 5 modified to accommodate MNC. The e_(MNC)[p_(t)] sequence (determined by FIGS. 11A-11C, discussed below) is subtracted from x_(f)[n_(t)] prior to the ΔΣ modulator, and the output range of the ΔΣ modulator 508, the range of the fast DEM encoder 510, and the number of FCEs of the FCE bank 502 driven by the fast DEM encoder 510 are all four times those of the original FIG. 5 system to accommodate the resulting dynamic range increase. Thus, f_(F)(t) is still given by (24), but now γ_(ΔΣ)[p_(t)] is given by the right side of (23) minus e_(MNC)[p_(t)]. Despite having the same qualitative properties as before, α_(F) and e_(F)(t) in (24) are slightly different in the modified system because of the additional FCEs.

An analysis shows that f_(time)(t) is now given by

f _(tune)(t)=α_(I)(t)a[g(w _(t))]+γ_(I)(t)d[g(w _(t)−1)]+α_(F) e _(ΔΣ)[w _(t)]+e _(R)(t),  (33)

where

e _(R)(t)=e _(M)(t)−α_(F) e _(MNC)[w _(t)]  (34)

is the residual FCE mismatch error, i.e., what is left of e_(M)(t) when e_(MNC)[p_(t)] is applied. It follows from (30), (32) and (34) that

$\begin{matrix} {{{e_{R}(t)} = {\Delta {\sum\limits_{k,r}^{\;}\left\{ {{\delta_{k,{r - {res}}}{s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} + {{\gamma_{k,{r - {res}}}(t)}\left( {{s_{k,r}\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack} - {s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} \right)}} \right\}}}},} & (35) \end{matrix}$

respectively.

Given that δ_(k,r) is constant, there exists an a_(k,r) that causes δ_(k,r-res)=0. In contrast, there is no b_(k,r) that causes γ_(k,r-res)(t) to vanish completely, because γ_(k,r)(t) is not constant. However, γ_(k,r)(t) is T_(fast)-periodic so there exists a b_(k,r) that makes the DC component of γ_(k,r-res)(t) zero, such that γ_(k,r-res)(t) is a linear combination of sinusoids with frequencies that are non-zero multiples of f_(fast). Therefore, it follows from Error! Reference source not found. that if

$\begin{matrix} {{a_{k,r} = {{\frac{\delta_{k,r}}{\alpha_{F}}\mspace{14mu} {and}\mspace{14mu} b_{k,r}} = {\frac{1}{\alpha_{F}T_{fast}}{\int_{0}^{T_{fast}}{{\gamma_{k,r}(\tau)}d\; \tau}}}}},} & (36) \end{matrix}$

for each k and r, then

δ_(k,r-res)=0 and ∫₀ ^(T) ^(fast) γ_(k,r-res)(τ)dτ=0.  (37)

In the absence of FCE static mismatches, a_(k,r)=0, and if the FCE frequency transitions are ideal, b_(k,r)=0.

Phase error is the integral of frequency error, so the DCO phase error introduced by e_(R)(t) is given by

θ_(R)(t)=∫₀ ^(t) e _(R)(τ)dτ.  (38)

If (37) is satisfied, then (35) and (38) imply that

$\begin{matrix} {{{\theta_{R}(t)} = {\Delta {\sum\limits_{k,r}^{\;}{\left( {{s_{k,r}\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack} - {s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} \right){\int_{0}^{t - {p_{t}T_{fast}}}{{\gamma_{k,{r - {res}}}(u)}{du}}}}}}},} & (39) \end{matrix}$

where t−p_(t)T_(fast)=t−└f_(fast)t┘T_(fast)<T_(fast). The term within the parenthesis in (39) equals zero when g(w_(t))−g(w_(t)−1)=0 and s_(k,r)[g(w_(t))−1] s_(k,r)[g(w_(t))] otherwise. Given that g(w_(t))−g(w_(t)−1) can only take on values from the set {0, 1}, then

s _(k,r)[g(w _(t)−1)]−s _(k,r)[g(w _(t))]=(g(w _(t))−g(w _(t)−1))(s _(k,r)[g(w _(t))−1]−s _(k,r)[g(w _(t))]).  (40)

Furthermore, g(w_(t)) is a T_(fast)-delayed version of n_(t), which increases by one unit every T_(in)=1/f_(in), so g(w_(t))−g(w_(t)−1) is T_(in)-periodic and is given by

$\begin{matrix} {{{g\left( w_{t} \right)} = {{g\left( {w_{t} - 1} \right)} = {\sum\limits_{k = {- \infty}}^{\infty}{r\left( {t - {kT}_{i\; n}} \right)}}}},} & (41) \end{matrix}$

where r(t)=1 for t∈[T_(fast), 2T_(fast)) and 0 otherwise. It follows from (41) that the Fourier expansion of g(w_(t))−g(w_(t)−1) is

$\begin{matrix} {\frac{f_{i\; n}}{f_{fast}} + {\sum\limits_{m = 1}^{\infty}{\frac{2}{m\; \pi}{\sin \left( {m\; \pi \frac{f_{i\; n}}{f_{fast}}} \right)}{{\cos \left( {2\pi \; {{mf}_{i\; n}\left\lbrack {t - {\frac{3}{2}T_{fast}}} \right\rbrack}} \right)}.}}}} & (42) \end{matrix}$

Thus, if the conditions shown in (37) are satisfied, (39), (40) and (42) imply that θ_(R)(t) would be given by second-order shaped noise multiplied by a T_(in)-periodic waveform and a DC-free T_(fast)-periodic waveform. Consequently, e_(R)(t) would introduce components with frequencies around f_(n,m)=nf_(fast)±mf_(in) to the DCO's phase error, where n=1, 2, 3, . . . and m=0, 1, 2, . . . . It follows from (42) that the power of the components around frequencies f_(n,m) with m near multiples of f_(fast)/f_(in) is very low. Therefore, θ_(R)(t) would not be a problem if f_(fast) is large enough because e_(R)(t) would only introduce high-frequency components to the DCO's phase error that would be lowpass filtered by the DCO. Simulation results also suggest that θ_(R)(t) is not a problem provided the conditions shown in (37) are satisfied and f_(fast) is large enough.

FCE Mismatch Error Measurement

The ideal MNC coefficient values are estimated as part of a feedback loop in a digital fractional-N PLL that incorporates the DCO. This is done during the PLL's normal operation by adaptively adjusting a_(k,r), and b_(k,r), such that the conditions shown in (37) are satisfied for each k and r, thereby minimizing e_(R)(t).

The purpose of a fractional-N PLL is to generate a periodic output signal, v_(PLL)(t), with frequency f_(PLL)=(N+α)f_(ref), where N is a positive integer, α is a fractional value and f_(ref) is the frequency of a reference oscillator waveform, v_(ref)(t). The general form of a digital fractional-N PLL without MNC is shown in FIG. 9. It consists of a phase-error-to-digital converter (PEDC) 902, a lowpass digital loop filter (DLF) 904, and a DCO 906. The PEDC's 902 output is an f_(ref)-rate digital sequence of the form

p[n]=−θ_(PLL)[n]+e _(p)[n],  (43)

where θ_(PLL)[n] is an estimate of the PLL's phase error and e_(p)[n] is additive error that includes quantization error from the PEDC's 902 digitization process as well as error from circuit noise and other non-ideal circuit behavior in both the PEDC and reference oscillator.

A modified version of the DCO 906 contains the preferred multi-rate DEM structure of FIG. 5 with the MNC correction of FIG. 8 with f_(in)=f_(ref). Typically, f_(fast)-clk is a divided-down version of v_(PLL)(t). Given that f_(PLL)=(N+α)f_(ref), f_(ref) and f_(fast) are incommensurate frequencies when α≠0, it is not possible for n_(t) to change synchronously with p_(t)=└f_(fast)t┘ if n_(t)=└f_(ref)t┘. Therefore, as shown in FIGS. 10A & 10B, in practice the DCO input is synchronized to f_(fast)-clk so (4) is satisfied, i.e., so n_(t) only changes at times μ_(n), which are multiples of T_(fast), instead of times nT_(ref), where T_(ref)=1/f_(ref) is the reference period. It is common practice in digital PLLs to synchronize the DLF 904 output to the clock signal of the fractional path, so this is not a special requirement of the proposed system. A conventional circuit to avoid metastability issues is also needed as part of the synchronization circuit shown in FIG. 10A, but it has been omitted for simplicity.

A requirement of a PLL is to suppress low-frequency DCO error, which is achieved by subjecting additive frequency error introduced by the DCO to a highpass filter that has at least one zero at DC. In the following, the impulse response of this filter is denoted as h[n], and its running sum, i.e., h[0]+h[1]+ . . . +h[n], is denoted as l[n]. p[n] can be written as

p[n]=p _(ideal)[n]+p _(R)[n],  (44)

where p_(ideal)[n] represents the contribution to p[n] of all noise sources except FCE mismatches and p_(R)[n] is the contribution to p[n] from e_(R)(t). Specifically, p_(R)[n] (with references to definitions in Appendix B) is given by

$\begin{matrix} {{{p_{R}\lbrack n\rbrack} = {{\Delta\alpha}_{F}T_{fast}{\sum\limits_{i = 0}^{n - 1}{\sum\limits_{k,r}^{\;}{\left\{ {{y_{k,{r - a}}\lbrack i\rbrack} + {y_{k,{r - b}}\lbrack i\rbrack}} \right\} {l\left\lbrack {n - 1 - i} \right\rbrack}}}}}},} & (45) \end{matrix}$

where y_(k,r-a)[t]+y_(k,r-b)[i] is proportional to the PLL's frequency error introduced by the s_(k,r)[n] sequences. If a_(k,r) and b_(k,r) in (32) are replaced by a_(k,r)[n_(t)] and b_(k,r)[n_(t)], respectively, then

y _(k,r-a)[i]=(q _(i−1)−3)s _(k,r)[i−1]a _(k,r-error)[i−1]+3s _(k,r)[i]a _(k,r-error)[i]  (46)

and

a _(k,r-b)[i]=(s _(k,r)[i−1]−s _(k,r)[i])b _(k,r-error)[i],  (47)

where q_(i−1) is the number of T_(fast) periods between times μ_(i−1) and μ_(i), and

a _(k,r-error)[n]=a _(k,r)[n]−a _(k,r) and b _(k,r-error)[n]=b _(k,r)[n]b _(k,r)   (48)

are the MNC coefficient errors at sample time n.

The term proportional to s_(k,r)[i] in (46) arises because the time at which the PEDC 902 samples the PLL's phase error, which is given by μ_(n)+4T_(fast) in the design example, is not equal to the time at which the integer FCE bank's inputs are updated, i.e., μ_(n)+T_(fast). Accordingly, the integer FCE bank's inputs are updated three T_(fast) before the PLL's phase error is sampled, which causes y_(k,r-a)[i] to depend on s_(k,r)[i−1] and also on s_(k,r)[i]. As implied by (44)-(47), the PEDC's 902 output has information regarding the MNC coefficient errors. The MNC coefficient estimation process described next is based on this result and on the properties of the switching sequences.

MNC Coefficients Estimation

A digital fractional-N PLL with the multi-rate DEM encoder and MNC technique is shown in FIG. 11A. The details of MNC logic 1102 are shown in FIG. 11B and FIG. 11C, wherein

$\begin{matrix} {{t_{k,r}\lbrack n\rbrack} = {\sum\limits_{i = 0}^{n}{s_{k,r}\lbrack i\rbrack}}} & (49) \end{matrix}$

is the running sum of s_(k,r)[n], and K_(a) and K_(b) are called the MNC gains. The MNC logic block consists of an adder and 25 s_(k,r)[n_(t)] residue estimators 1104.

It follows from FIG. 4 that each s_(k,r)[n] sequence is a concatenation of sequences of the form 1, 0, . . . , 0, −1, 0, . . . , 0 or −1, 0, . . . , 0, 1, 0, . . . , 0, where each 0 is present only when the input of the s_(k,r)[n] generator is zero. Thus, |s_(k,r)[n]|≤1, |s_(k,r)[n]|≤1 and |s_(k,r)[n]−s_(k,r)[n−1]|≤2 for all n, so the multipliers in FIG. 11(c) are simple in terms of hardware.

The s_(k,r)[n_(t)] residue estimators 1104 are responsible for the computation of the MNC coefficients. At each sample time, the MNC coefficient errors are measured and a_(k,r)[n_(t)] and b_(k,r)[n_(t)] are updated such that they approach the values shown in (36). The measurement of the MNC coefficient errors is based on the statistical properties of the switching sequences.

Although each s_(k,r)[n] sequence depends on the input of its corresponding SB, when it is non-zero, its sign depends on d_(k,r)[n]. Given that the d_(k,r)[n] sequences are independent of the d_(k,r)[n] sequences in the other SBs, this provides enough randomization for the s_(k,r)[n] sequences to be uncorrelated with each other. Furthermore, as the d_(k,r)[n] sequences are also independent of all electronic device noise sources in the PLL, each s_(k,r)[n] sequence is uncorrelated with all such sources as well, and it is also uncorrelated with the PEDC's 902 quantization noise in PLLs where such noise source is uncorrelated with the PLL's phase error.

Hence, in such cases, the s_(k,r)[n] sequences are uncorrelated with all PLL noise except the terms in p[n] arising from e_(R)(t), i.e., p_(R)[n].

The y_(k,r-a)[i] and y_(k,r-b)[i] terms in p[n] depend on the MNC coefficient errors, and such terms are proportional to functions of the s_(k,r)[n] sequences. Specifically, it can be seen from (44)-(47) that p[n] has information about an accumulated version of

(q _(n−2)−3)s _(k,r)[n−2]a _(k,r-error)[n−2],  (50)

and that p[n]−p[n−1] has information about

(s _(k,r)[n−2]s _(k,r)[n−1])b _(k,r-error)[n−1].  (51)

Therefore, it follows that the accumulator inputs in FIG. 11C, i.e., −p[n]t_(k,r)[n−2] and (p[n−1]−p[n])(s_(k,r)[n−2]−s_(k,r)[n−1]), when non-zero, are noisy estimates of a_(k,r-error)[n] and b_(k,r-error)[n], respectively, so they can be used to adaptively compute the ideal MNC coefficients. In practice, the top and bottom branches within each s_(k,r)[n_(t)] residue estimator 1104 interfere with each other in a way that makes the accumulator inputs have information about both MNC coefficient errors. However, extensive simulations indicate that the MNC coefficient values converge to their ideal values regardless of such interferences provided the MNC gains are set properly to avoid instability in the feedback loops.

It would also be possible to correlate p[n−1]−p[n] by s_(k,r)[n−2] to get an estimate of a_(k,r-error)[n]. However, as a_(k,r)[n] is only updated when the accumulator input is non-zero, correlating p[n−1]−p[n] against s_(k,r)[n−2] instead of −p[n] against t_(k,r)[n−2] would significantly decrease the convergence speed of a_(k,r)[n] because normally s_(k,r)[n−2] is zero more often than t_(k,r)[n−2]. Although correlating −p[n] against t_(k,r)[n−2] effectively increases the error variance of a_(k,r)[n], as explained next, this problem can be mitigated by reducing K_(a).

As is common in most LMS-like algorithms, the choice of K_(a) and K_(b) represents a tradeoff. The larger the MNC gains, the faster the convergence, but the larger the error variance of a_(k,r)[n] and b_(k,r)[n]. Also, as the s_(k,r)[n_(t)] residue estimators comprise two LMS-like loops in parallel that interfere with each other, K_(a) and K_(b) each affect the convergence time and error variance of both a_(k,r)[n] and b_(k,r)[n]. Although it might be possible to develop closed-form expressions that quantify these tradeoffs, the authors currently use simulations to assist the design process and to choose the values of K_(a) and K_(b).

Simulation Results

The multi-rate DEM and the MNC methods of the preferred embodiments described above were tested in an event-driven behavioral simulation of a modified version of the ΔΣ frequency-to-digital converter based fractional-N PLL presented in [C. Weltin-Wu, G. Zhao, and I. Galton, “A Highly-Digital Frequency Synthesizer Using Ring-Oscillator Frequency-to-Digital Conversion and Noise Cancellation,” IEEE International Solid-State Circuits Conf., pp. 1-3, February 2015; C. Weltin-Wu, G. Zhao, and I. Galton, “A 3.5 GHz Digital Fractional-N Frequency Synthesizer Based on Ring Oscillator Frequency-to-Digital Conversion,” IEEE J. Solid-State Circuits, vol. 50, no. 12, pp. 2988-3002, December 2015]. As explained in [C. Weltin-Wu, E. Familier, and I. Galton, “A Linearized Model for the Design of Fractional-N PLLs based on Dual-Mode Ring Oscillator FDCs,”IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 8, pp. 2013-2023, August 2015], p[n] is given by (43) where e_(p)[n] is first-order shaped quantization noise that is uncorrelated with the PLL's phase error plus error from both the PEDC and reference oscillator.

The DLF consists of two single-pole IIR stages and a proportional-integral stage. Its transfer function is

$\begin{matrix} {{{L(z)} = {{K_{M}\left( {K_{P} + \frac{K_{I}}{1 - z^{- 1}}} \right)}{\prod\limits_{i = 0}^{1}\frac{\lambda_{i}}{1 - {\left( {1 - \lambda_{i}} \right)z^{- 1}}}}}},} & (52) \end{matrix}$

where K_(M), K_(P), K_(I), λ₀ and λ₁ are constant loop filter parameters. The DCO consists of an LC oscillator core with a power-of-two-weighted coarse capacitor bank, an integer FCE bank 502 and a fractional FCE bank 504 in accordance with FIG. 5. The latter two are driven by the multi-rate DEM encoder 500 shown in FIG. 5 and modified as shown in FIG. 8 with f_(fast)=f_(PLL)/8 and Δ_(min)=40 kHz (i.e., Δ=156.25 Hz).

The static gain error of the ith FCE was modeled as an additive zero-mean Gaussian random variable with a standard deviation of 5% of Δ_(i) divided by the square root of Δ_(i)/Δ_(min). The FCE frequency transitions were modeled as second-order transients that settle within one T_(fast) period. The parameters of these transients, such as the damping factor and the natural frequency, are modelled as random variables with means and standard deviations determined from transistor-level simulation results. FIG. 12 shows example frequency transients used in the simulation.

The simulated noise parameters of the DCO and the reference oscillator, as well as the PEDC internal parameters were f_(ref)=26 MHz, N=134 and α=0.0003846153, so that f_(PLL)=3.484 GHz and f_(fast)=435.5 MHz. The DLF parameters used were K_(M)=1.25, K_(P)=2⁴, K_(I)=2⁻⁴, λ₀=2⁻³ and λ₁=2⁻², and the MNC gains were set to K_(a)=2⁻³ and K_(b)=2⁻⁵. The simulated PLL has a bandwidth of 206 kHz and a phase margin of 63 degrees.

FIG. 13A shows the simulated PLL phase noise PSD with the multi-rate DEM technique disabled, i.e., with the flip-flops in both the slow and fast DEM encoders frozen. The two curves in FIG. 13A were obtained from two different simulations: one in which d_(I)[n_(t)] is constant and another one in which d_(I)[n_(t)] changes frequently. Although the DCO input sequence does not vary significantly in the short term once the PLL is locked, its moving average drifts over time such that d_(I)[n_(t)] eventually begins to change frequently, at which point it degrades the PLL's phase noise as shown in FIG. 13A. Once the multi-rate DEM technique is enabled, whether or not d_(I)[n_(t)] changes has no significant effect on the DCO's frequency, so spectral breathing no longer occurs.

FIG. 13B shows the simulated PLL phase noise PSD with the multi-rate DEM technique enabled for two cases: one case with just static gain errors, and the other case with just non-ideal frequency transitions. FIG. 13C shows the simulated PLL phase noise PSD considering both sources of error with the multi-rate DEM technique enabled and with the MNC technique disabled and enabled. The theoretical PLL phase noise PSD for ideal FCEs, which was computed using the linearized model presented in [C. Weltin-Wu, E. Familier, and I. Galton, “A Linearized Model for the Design of Fractional-N PLLs based on Dual-Mode Ring Oscillator FDCs,”IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 62, no. 8, pp. 2013-2023, August 2015], is also plotted as the dashed curves in FIGS. 13A-13C to provide a comparison baseline.

As shown in FIG. 13A, when the MNC technique is enabled the resulting phase noise PSD matches the theoretically-predicted phase noise PSD for ideal FCEs after 13·10⁷ reference periods (5 seconds) from a cold start. This implies a phase noise improvement of more than 20 dB at an offset frequency around 10 MHz. As the FCE mismatches are mostly determined by circuit component mismatches, they are not expected to change significantly over time. Hence, once obtained, the MNC coefficients can be stored in memory and used subsequently by the PLL, thereby avoiding future convergence time delays.

FIG. 14 shows the evolution of the MNC coefficient errors over time from the simulation used to generate the curves in FIG. 13C. As shown in FIG. 14, some b_(k,r)[n] coefficients initially move away from their ideal values. As explained above, this happens because the top and bottom branches of each s_(k,r)[n_(t)] residue estimator interfere with each other so that the error estimate at the input of each accumulator is biased by the MNC coefficient error of the opposite branch. If the MNC gains are set as in FIGS. 14A and 14B, this is not a problem because this effect becomes less significant as either one or both MNC coefficients approach their ideal values.

It follows from (46) and (47) that the terms proportional to a_(k,r-error)[n] in p[n] are q_(n)−3 times larger than those proportional to b_(k,r-error)[n] (e.g., q_(n)≅16 in the design example), so for K_(a)=K_(b), the error variance of each b_(k,r)[n] is expected to be larger than that of a_(k,r)[n]. Therefore, in order to make the error variance of the b_(k,r)[n] coefficients comparable to that of the a_(k,r)[n] coefficients, K_(b) has to be smaller than K_(a). As shown in FIGS. 14A & 14B, this causes the b_(k,r)[n] coefficients to converge to their ideal values at a slower rate than the a_(k,r)[n] coefficients, so the convergence speed of the MNC technique is limited by K_(b). Nonetheless, it follows from FIGS. 14A & 14B that the a_(k,r)[n] coefficients get close to their ideal values in less than 10⁷ reference periods (˜0.4 seconds). Hence, as the most significant sources of phase noise are the FCE static gain errors, the MNC method allows for a considerable phase noise improvement in less than half a second.

To reduce the cold-start convergence time of the MNC technique, large MNC gains can be used initially and decreased over time. See, W. Y. Chen and R. A. Haddad, “A Variable Step Size LMS Algorithm,” Proc. 33^(rd) Midwest Symp. Circuits and Systems, pp. 423-426, August 1990. FIGS. 15A and 15B shows the evolution of the MNC coefficient errors over time for 7.8.10⁷ reference periods (3 seconds) for an example case in which K_(a) and K_(b) are initially set to 2⁻¹ and 2⁻², respectively, and then divided by two at the times indicated by the vertical dashed lines. In this case, the MNC coefficients reach the final values shown in FIGS. 14A & 14B in roughly 3 seconds, and the a_(k,r)[n] coefficients get close to their ideal values in less than 2·10⁶ reference periods (˜0.08 seconds), which is five times faster than in FIGS. 14A & 14B.

Appendix A

It follows from FIG. 5 and (17) that

$\begin{matrix} {{f_{I}(t)} = {\sum\limits_{i = 5}^{22}{\left\lbrack {{\left( {{b_{i}\left\lbrack w_{t} \right\rbrack} - {1/2}} \right){\alpha_{i}(t)}\Delta_{i}} + {\left( {{b_{i}\left\lbrack {w_{t} - 1} \right\rbrack} - {1/2}} \right){\gamma_{i}(t)}}} \right\rbrack.}}} & (53) \end{matrix}$

Expressions for each b_(i)[w_(t)]=c_(i+12)[g(w_(t))] in terms of d[g(w_(t))] and the switching sequences can be found by tracing through the tree of FIG. 6 and applying (20) and the expressions shown in FIG. 4(a) and FIG. 4(b). This leads to

$\begin{matrix} {{{{c_{i}\left\lbrack {g\left( w_{t} \right)} \right\rbrack} - {1/2}} = {{m_{i}{{d\left\lbrack {g\left( w_{t} \right)} \right\rbrack}/\Delta}} + {\sum\limits_{k,r}^{\;}{\kappa_{k,r,i}{s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}}}}},} & (54) \end{matrix}$

where

m _(i)=0 for 17≤i≤26 and m _(i)=2⁻¹⁶ for 27≤i≤34,  (55)

and each x_(k,r,i) is one of 0, −½, ½, 2^(−k) or 2^(−k). Combining (4)(19) and (53)-(55) yields (26) and (27), where α_(I)(t) and γ_(I)(t) are the averages of α_(i)(t) and (2⁻¹³/Δ)γ_(i)(t) for i=15, 16, . . . , 22, respectively,

$\begin{matrix} {{\alpha_{k,r}(t)} = {{\sum\limits_{i = 5}^{22}{{\alpha_{i}(t)}K_{i + 12}\kappa_{k,r,{i + 12}}\mspace{14mu} {and}\mspace{14mu} {\gamma_{k,r}(t)}}} = {\sum\limits_{i = 5}^{22}{\frac{\gamma_{i}(t)}{\Delta}{\kappa_{k,r,{i + 12}}.}}}}} & (56) \end{matrix}$

Each α_(I)(t), γ_(I)(t), α_(k,r)(t) and γ_(k,r)(t) is T_(fast)-periodic, because it is a linear combination of α_(I)(t) and γ_(i)(t), which are T_(fast)-periodic.

Appendix B

The phase error of the digital PLL shown of FIG. 9 is given by

θ_(PLL)(t)=∫₀ ^(t) ψv _(PLL)(u)du,  (57)

where ψ_(PLL)(t) is the PLL's frequency error at time t. The θ_(PLL)[n] term in (43) is a sampled version of θ_(PLL)(t) given by

θ_(PLL)[n]=θ_(PLL)(τ_(n)),  (58)

where τ_(n)=nT_(ref)+ and λ_(n) is a small implementation-dependent deviation of τ_(n) from its ideal value. It follows from (43), (57) and (58) that

$\begin{matrix} {{{p\lbrack n\rbrack} = {{p\lbrack 0\rbrack} - {T_{ref}{\sum\limits_{i = 1}^{n}{\psi_{PLL}\lbrack i\rbrack}}} + {e_{p}\lbrack n\rbrack}}},} & (59) \end{matrix}$

where

$\begin{matrix} {{\psi_{PLL}\lbrack i\rbrack} = {\frac{1}{T_{ref}}{\int_{\tau_{i - 1}}^{\tau_{i}}{{\psi_{PLL}(u)}{du}}}}} & (60) \end{matrix}$

is the PLL's average frequency error over the time interval [τ_(i,1), τ_(i)] and p[0] is the initial value of p[n]. FIG. 9 and (60) imply that e_(R)(t) causes a term in ψ_(PLL)[i] given by

$\begin{matrix} {{{\left\{ {e_{R}*h} \right\} \lbrack i\rbrack} = {\sum\limits_{j = 0}^{\infty}{{h\lbrack j\rbrack}{e_{R}\left\lbrack {i - j} \right\rbrack}}}},} & (61) \\ {where} & \; \\ {{e_{R}\lbrack i\rbrack} = {\frac{1}{T_{ref}}{\int_{\tau_{i - 1}}^{\tau_{i}}{{e_{R}(u)}{du}}}}} & (62) \end{matrix}$

and h[j] is the impulse response of the highpass filtering operation imposed by the PLL on the DCO's additive frequency error as discussed in the description above.

In the design example of the example embodiment λ_(n)=4.2T_(fast)+⅛T_(fast)v[n], where v[n] is an integer-valued sequence restricted to the set {−6, 5, . . . , 5, 6}, so τ_(n)=nT_(ref)+4.2T_(fast)+⅛T_(fast)v[n]. As the magnitude of ⅛T_(fast)v[n] is at most ¾T_(fast), its effect is negligible. Furthermore, for the sake of simplicity, τ_(n) is assumed to be given by

τ_(n)=μ_(n)+4T _(fast),  (63)

where μ_(n), as shown FIG. 10B, is a multiple of T_(fast). Given that 0<μ_(n)−nT_(ref)≤T_(fast) for all n and that T_(fast) is a small fraction of T_(ref), this approximation does not significantly affect the following results. Substituting Error! Reference source not found. with a_(k,r) and b_(k,r) replaced by a_(k,r)[g(w_(t))] and b_(k,r)[g(w_(t))], respectively, into (35), and the result of this operation and (63) into (62) yields

$\begin{matrix} {{e_{R}\lbrack i\rbrack} = {\frac{\Delta}{T_{ref}}{\int_{\mu_{i - 1} + {4T_{fast}}}^{\mu_{i} + {4T_{fast}}}{\left\{ {{\left( {\delta_{k,r} - {\alpha_{F}{a_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}}} \right){s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} + {\left( {{\gamma_{k,r}(t)} - {\alpha_{F}{b_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}}} \right)\left( {{s_{k,r}\left\lbrack {g\left( {w_{t} - 1} \right)} \right\rbrack} - {s_{k,r}\left\lbrack {g\left( w_{t} \right)} \right\rbrack}} \right)}} \right\} {{dt}.}}}}} & (64) \end{matrix}$

Given that t∈[μ_(n), μ_(n+1)) implies g(p_(t))=n−1, it follows that g(w_(t))=i−2 for t∈[μ_(i−1)+T_(fast), μ_(i)+T_(fast)) and g(w_(t))=i−1 for t∈[μ_(i)+T_(fast), μ_(i)+T_(fast)), so (64) can be written as

$\begin{matrix} {{{e_{R}\lbrack i\rbrack} = {{- \Delta}\frac{\alpha_{F}T_{fast}}{T_{ref}}{\sum\limits_{k,r}^{\;}\left\{ {{\gamma_{k,{r - a}}\left\lbrack {i - 1} \right\rbrack} + {y_{k,{r - b}}\left\lbrack {i - 1} \right\rbrack}} \right\}}}},} & (65) \end{matrix}$

where y_(k,r-a)[i] and y_(k,r-b)[i] are given by (46) and (47), respectively, and it has been assumed that q_(i)=(μ_(i+1)−μ_(i))/T_(fast) is greater than 3 for all i (e.g., q_(i)≅16 in the design example). Substituting (65) into (61) and the result into (59), rearranging terms and considering that s_(k,r)[n]=0 for n<0 gives (44) and (45)

While specific embodiments of the present invention have been shown and described, it should be understood that other modifications, substitutions and alternatives are apparent to one of ordinary skill in the art. Such modifications, substitutions and alternatives can be made without departing from the spirit and scope of the invention, which should be determined from the appended claims.

Various features of the invention are set forth in the appended claims. 

1. A digital fractional-N phase locked loop (PLL) with multi-rate dynamic element matching (DEM) and an adaptive mismatch-noise cancellation (MNC), comprising: a phase error to digital converter; a digital loop filter to suppress quantization noise of the phase error to digital converter and drive a digitally controlled oscillator; a digitally controlled oscillator (DCO) with a multi-rate DEM encoder driving an integer bank of frequency control elements and a fractional bank of frequency control elements; and adaptive mismatch-noise cancellation logic operating to cancel DCO phase error arising from frequency control element (FCE) static and dynamic mismatch error by estimating ideal MNC coefficient values during PLL normal operation, estimating MNC coefficient errors at each sample time, and updating the MNC coefficient values to approach zero FCE static and dynamic mismatch error.
 2. The digital fractional-N phase locked loop of claim 1, wherein the updating of the MNC coefficient values is conducted once for each time the phase error of the PLL is measured.
 3. The digital fractional-N phase locked loop of claim 1, wherein the multi-rate DEM comprises: a slow DEM encoder that drives the integer bank of frequency control elements and a second order ΔΣ modulator; and a fractional path, wherein the fractional path includes the second-order digital ΔΣ modulator driving a fast DEM encoder that drives the fractional bank of frequency control elements, wherein the second-order digital ΔΣ modulator and fast DEM encoder are clocked at a higher frequency compared to that of the slow DEM encoder.
 4. The digital fractional-N phase locked loop of claim 3, wherein the ΔΣ modulator's quantization noise is asymptotically independent of its input and dither sequences used in the ΔΣ modulator.
 5. The digital fractional-N phase locked loop of claim 3, wherein the adaptive mismatch-noise cancellation logic injects an MNC correction sequence, which is computed from the MNC coefficient values and the switching sequences generated inside the slow DEM encoder, into the fractional path.
 6. The digital fractional-N phase locked loop of claim 3, wherein the adaptive mismatch-noise cancellation logic estimates the ideal MNC coefficients with a least-mean-square (LMS)-like algorithm.
 7. The digital fractional-N phase locked loop of claim 3, wherein the adaptive mismatch-noise cancellation logic estimates the ideal MNC coefficients based on the statistical properties of switching sequences generated inside the slow DEM encoder. 